A308523 Number of essentially simple rooted toroidal triangulations with n vertices.
0, 1, 10, 97, 932, 8916, 85090, 810846, 7719048, 73431340, 698187400, 6635738209, 63047912372, 598885073788, 5687581936284, 54005562798252, 512728901004816, 4867263839614716, 46199494669833400, 438481077306427924, 4161316466910824272
Offset: 0
Keywords
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..499
- Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, A bijection for essentially 3-connected toroidal maps, arXiv:1907.04016 [math.CO], 2019.
- Éric Fusy, Benjamin Lévêque, Orientations and bijections for toroidal maps with prescribed face-degrees and essential girth, arXiv:1807.00522 [math.CO], 2018. See Proposition 25 p. 37.
Crossrefs
Programs
-
Maple
n:=20: dev_A := series(RootOf(A-x*(1+A)^4, A), x = 0, n+1): seq(coeff(series(subs(A=dev_A, A/(1-3*A)^2), x, n+1), x, k), k=0..n);
-
Mathematica
terms = 21; A[] = 0; Do[A[x] = x (1 + A[x])^4 + O[x]^terms, terms]; CoefficientList[A[x]/(1 - 3 A[x])^2, x] (* Jean-François Alcover, Jun 17 2019 *)
-
PARI
my(N=30, x='x+O('x^N), g=x*sum(k=0, N, binomial(4*k+2, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-4*g)^2)) \\ Seiichi Manyama, Jul 19 2025
Formula
G.f.: A/(1-3*A)^2 where A=x(1+A)^4 is the g.f. of A002293.
From Vaclav Kotesovec, Jun 25 2019: (Start)
Recurrence: 81*(n-1)*(3*n - 2)*(3*n - 1)*(24*n - 37)*a(n) = 24*(13824*n^4 - 59328*n^3 + 92832*n^2 - 62278*n + 14653)*a(n-1) - 2048*(2*n - 3)*(4*n - 7)*(4*n - 5)*(24*n - 13)*a(n-2).
a(n) ~ 2^(8*n - 3) / 3^(3*n). (End)
From Seiichi Manyama, Jul 19 2025: (Start)
G.f.: g*(1-g)/(1-4*g)^2 where g*(1-g)^3 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/4) * log( Sum_{k>=0} binomial(4*k,k)*x^k ). (End)
From Seiichi Manyama, Jul 28 2025: (Start)
a(n) = Sum_{k=0..n-1} binomial(4*k-2+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n-1,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k-1,k). (End)